A Dynamic Model of Conflict and Cooperation

because the value function (C.2) associated with this strategy is unbounded at Z = n∕δ and
thus the strategy a(Z)=0for Z ∈ [0, ∞) ceases to be continuous at this point.

Next, consider the case where c2 =0. In this case it turns out that the derivative of the
resulting value function with respect to Z is bounded above:

On other hand, since

lim ^dpⁱZ = lim r ⁽n- 1) Z → ∞ for ∀Z > 0,                 (C.4)

ai→0∂a_i     ai→0 n2a

inequality (∂pi/dai) Z < V0 (Z) never holds except for Z = 0. Hence, the cornered strategy
α(Z) = 0 is not an equilibrium strategy for Z ∈ (0, ∞). ■

Lemma 2 There exists a constant of integration which makes the strategy a(Z)=1 an equi-
librium strategy over Z ∈ (0, ∞).

Proof. When all players play strategy a_i = 1, the HJB equation (6) becomes

ρV (Z) = nZ + V' (Z)(-δZ).

By integration and imposing symmetry, we have

where c₃ represents a constant of integration. When setting a = 1 in ∂p_i∕∂a_i yields (∂p_i∕∂a_i)Z ≡
r (n — 1) Z/n², the first-order condition (∂p_i∕∂a_i) Z ≥ V0 (Z) only allows (C.6) to hold for the
values of Z satisfying

δ 1 .+’ r (n — 1) -, 2'
— ----Z ^δ--------Z ^δ

ρn ρ + δ         n

Since the first exponent inside the brackets on the left-hand side of (C.7) is smaller than the
second exponent term, it dominates for smaller values of Z, whereas for larger values of Z

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